Mathematical Modeling and Solution Strategies for Nonlinear Differential Equations Using Advanced Theorems
Keywords:
Nonlinear differential equations, Ordinary differential equations (ODEs), Existence and uniqueness theorems, Picard-Lindelöf theorem, Peano existence theorem, Lyapunov stability, Hartman-Grobman theorem, Center manifold theoryAbstract
Nonlinear differential equations arise in diverse scientific and engineering fields, from ecology to physics. Unlike linear systems, they can exhibit complex dynamics such as multiple equilibria, limit cycles, and chaos. This paper provides a comprehensive survey of modeling and solution approaches for nonlinear ODEs. We discuss fundamental existence and uniqueness theorems (e.g. Picard-Lindelöf and Peano), and illustrate classic nonlinear models (logistic growth, predator-prey, epidemic models). We review analytical and numerical solution techniques, and delve into advanced theoretical tools: stability definitions (Lyapunov stability, Hartman-Grobman linearization), bifurcation theory (Hopf bifurcation), and invariant-manifold theorems (center manifold reduction). Throughout, we include examples, figures (phase portraits), tables of model equations, and code snippets for simulations. Detailed citations and references are provided.
References
1. Coddington, E. A., & Levinson, N. (1955). Theory of ordinary differential equations. New York: McGraw-Hill.
2. Gilbert, I. A., et al. (2016). Calculus: Single Variable. OpenStax. (Used for logistic growth model).
3. Hartman, P. (1964). Ordinary Differential Equations. New York: John Wiley & Sons.
4. Khalil, H. K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall.
5. Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A, 115(772), 700-721.
6. Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci., 20(2), 130-141.
7. Murray, J. D. (2002). Mathematical Biology I: An Introduction (3rd ed.). New York: Springer.
8. Poincaré, H. (1890). Sur les courbes définies par une équation différentielle. Paris: Gauthier-Villars.
9. Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, and engineering (2nd ed.). Westview Press.
10. Verhulst, P.-F. (1845). Recherches mathématiques sur la loi d’accroissement de la population. Nouvelle Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18, 1-41
11. Mustafa M. Abuali, & Tarek M. Ghomeed. (2024). Utilizing the Artificial Bee Algorithm to Enhance the Accuracy of Book Recommendation Systems: A Case Study on Goodreads dataset. Libyan Journal of Contemporary Academic Studies, 2(2), 54-62.
